How to convert from vertex_color to material and export FBX file
Unity is a game engine, not a vertex editing engine. While you can build or download tools to edit meshes inside of Unity, I'd advise you to stick to Blender for that. Unity displays everything similar to Eevee. If you assign a default material to an object in Eevee it also won't show vertex colors, unless you tell your material to read them. In Blender, you need a built-in node for that, in Unity you need a shader that can read and display them.
https://forum.unity.com/threads/finding-and-using-a-vertex-colors-shader.236366/
Unity will also omit any vertex color information in the build, unless you tell it to keep them.

This option is in the Player Settings.